welcome! jeremy freese is a professor in sociology at northwestern university. he finds blogging to be a good diversion from insomnia and a far better use of time than television.

Saturday, May 19, 2007

nonremarkable feats of saloon proving!

All I wanted to do was sit in the bar and watch the Spurs-Suns game in peace. But, somehow, I came to be challenged as a matter of honor to prove that the square root of two could not be expressed as a fraction. Indeed, the whole honor of sociology was said to hang on my ability to do this. Without using any paper. (Actually, I'm not sure what I would have done with paper anyway, so the grander gesture seemed to assert that paper was for pantywaists.)

This involved firing up some rather long dormant mental machinery and saying "uhhh" for protracted periods, thus being unable to concentrate on rooting for the Suns and thereby possibly contributing to their loss. But, to a level of astonishment that even caused the game to pause and Amare Stoudemire to look through the screen at me and say "Whoa!", I did it!

I will delete the various durations of inarticulate hemming that came between each step:

"So, if the square root of two was a fraction, it would be equal to x/y where both x and y would be integers."

"That would mean that the square of x was twice the square of y."

"So what I need to do is prove that a number that is twice the square of an integer can't be the square of another integer."

"Well, any integer can be expressed as its prime factors, and the prime factors of the square of an integer are just going to be those prime factors twice. So the square of an integer always has an even number of prime factors"

"Twice the square of an integer has to have an odd number of prime factors. Because its prime factors are going to be the prime factors of the square of the integer--an even number--and two."

"If twice the square of an integer has an odd number of prime factors, then it isn't itself the square of any integer. So there are no integers x and y for which the square of x is twice the square of y. Q-E-[expletive deleted]-D."

6 comments:

Nice work! Rumor has it that Hippasus came up with this very proof back in the day (I think around 500BC). Hippassus was a student of Pythagoras. The Pythagorians believed in the absolute purity of numbers. And so with this proof Hippasus showed them to be wrong. But the Pythaorians showed him in the end: they killed him for it. Or so goes the rumor.

The issue came up because we knew that the proof that the square root of two was an irrational number was how irrational numbers were discovered, but then came the question of how one would prove that. So I knew one had to be able to get there via nonarcane reasoning.